1、Attitude & Orbit Control System (AOCS) Introduction,Huaizu You National Space Organization12 April 2007,References,Peter C. Hughes, “Spacecraft Attitude Dynamics,” John Wiley & Sons, New York, 1986. Vladimir A. Chobotov, “Spacecraft Attitude Dynamics and Control,” Kreiger Publishing Company, Malabar
2、, Florida, 1991. James R. Wertz, “Spacecraft Attitude Determination and Control,” Microcosm Inc., Kluwer Academic Publishers, Norwell, Massachusetts, 1990.,Outline,Attitude/Orbit equations of motion AOCS hardware Attitude determination Attitude control AOCS design,Orbit (translation): Simple model:
3、Kepler time equation Complicated model: Lagrange planetary equation Newton 2nd law of motion Attitude (rotation): Kinematic eq. Dynamic eq.: Euler rotational eq. of motion,Attitude/Orbit equations of motion,Orbital dynamics: two-body problem,Two-body problem: (point masses) Conservation of energy (d
4、ot product with .) Conservation of angular momentum (cross product with ) Keplers 1st law: the orbit of each planet around the sun is an ellipse, with the sun at one focus. Keplers 2nd law: the radius vector from the sun to a planet sweeps out equal areas in equal time intervals. Keplers 3rd law: th
5、e square of the orbital period of a planet is proportional to the cube of the semi-major axis of the ellipse.,Orbital dynamics: three-body problem,Circular restricted three-body problem: the motion of the two primary bodies is constrained to circular orbits about their barycenter. Sun-Earth-Moon Lag
6、rangian (or libration) points Halo orbit (closed Lissajous trajectory: quasi-periodic orbit) Elliptic restricted three-body problem Earth-Moon-Satellite,Orbital dynamics: Keplers time eq.,Keplers time eq.: find the position in an orbit as a function of time or vice versa. Applicable not only to elli
7、ptic orbits, but all conic section families (parabola, hyperbola),M: mean anomaly, E: eccentric anomaly e: eccentricity, a: semimajor axis,Orbital dynamics: orbital elements,At a given time, we need 6 variables to describe the state in 3D translational motion (position+velocity) 6 orbital elements:
8、Semimajor-axis, a Eccentricity, e Inclination, i Right ascension of ascending node, W Argument of perigee, w Mean anomaly, M,Orbital dynamics: environmental perturbations,Conservative forces: Asphericity of the Earth: zonal/tesseral harmonics Third body gravitational field: Sun/Moon Non-conservative
9、 forces: Aerodrag: area-to-mass ratio Solar wind,Orbital dynamics: Lagrange planetary equation,Variation of parameters in ODE Singular at circular (eccentricity = 0) or stationary orbits (inclination = 0) equinoctial orbital elements Application: sun-synchronous orbit,Orbital Maneuvers,Launch vehicl
10、e trajectories: Vertical flight First-stage powered flight First-stage separation Second-stage powered flight Second-stage separation Coasting flight Third-stage powered flight Orbit injection Orbit injection Single-impulse maneuvers Hohmann transfer: two-impulse elliptic transfer (fuel optimal amon
11、g two-impulse maneuvers between two coplanar circular orbits) Interplanetary flight: 1) Earth escape, 2) heliocentric orbital transfer, and 3) planet encounter Orbital rendezvous: Clohessy-Wiltshire (or Hills) eq.,Attitude dynamics: rotational kinematics,Direction cosine matrix Eulers angles Eulers
12、eigenaxis rotation: space-axis and body-axis rotation Quaternions (or Euler parameters) Kinematic differential equations,Rotational kinematics: direction cosine matrix (orthonormal),Two reference frames with a right-hand set of three orthogonal bases:,Rotational kinematics: Eulers angles,Body-axis/s
13、pace-axis rotation: successively rotating three times about the axes of the rotated, body-fixed/inertial reference frame. 1) any axis; 2) about either of the two axes not used for the 1st rotation; 3) about either of the two axes not used for the 2nd rotation. each has 12 sets of Euler angles. Ex:,R
14、otational kinematics: Eulers eigenaxis rotation,Eulers eigenaxis rotation: by rotating a rigid body about an axis that is fixed to the body and stationary in an inertial reference frame, the rigid-body attitude can be changed from any given orientation to any other orientation.,Rotational kinematics
15、: quaternions,Euler parameters (quaternions): Why quaternions? Quaternions have no inherent geometric singularity as do Euler angles. Moreover, quaternions are well suited for onboard real-time computer because only products and no trigonometric relations exist in the quaternion kinematic differenti
16、al equations.,Rotational kinematics: kinematic differential equations,Reference: Kane, T.R., Likins, P.W., and Levinson, D.A., “Spacecraft Dynamics,” McGraw-Hill, New York, 1983.,Angular momentum of a rigid body The rotational eq. of motion of a rigid body about an arbitrary point O is given as The
17、absolute angular moment about point O is defined asEulers rotational equations of motion,Attitude dynamics: rigid-body dynamics,if at the center of mass,if it is a rigid body,J: moment of inertia matrix of a rigid body about a body-fixed reference frame with its origin at the center of mass.,Attitud
18、e dynamics: general torque-free motion (M=0),Angular velocity vector must lie on 1) angular momentum ellipsoid, and 2) kinetic energy ellipsoid at the same time intersection: polhode (seen from body-fixed reference frame) Analytical closed-form solution to the torque-free motion of an asymmetric rig
19、id body is expressed in terms of Jacobi elliptic functions. Stability of torque-free motion about principal axes: 1) major axis: stable; 2) intermediate axis: unstable; 3) minor axis: stable only if no energy dissipation.,Attitude dynamics: constant body-fixed torque (M=const.),Spinning axisymmetric
20、 body Possesses a gyrostatic stiffness to external disturbances (e.g., football) The path of the tip of the axis of symmetry in space is an epicycloid. Asymmetric rigid body About major or minor axis About intermediate axis,Gravitational Orbit-Attitude Coupling,Why coupling? Because the rigid body i
21、s not a point mass. Derivation: expand the Earth gravitational force in terms of Legendre polynomial, then the corresponding torque appears in higher order terms of the moment of inertia dyadic. Significant when the characteristic size of the satellite is larger than 22 km. Conclusion: dont worry ab
22、out this effect now.,Orbit: Newtons 2nd law of motionAttitude a): kinematic equationAttitude b): dynamic equation,Solve Attitude/Orbit dynamics numerically,Cowells formulation (Enckes method),AOCS hardware,Sensors: Sun sensor Magnetometer (MAG) Star tracker Gyro GPS receiver Actuators: Magnetorquer
23、(torque rod) (MTQ) Reaction wheel (RW) Thruster,Comparison of attitude sensors,courtesy from Oliver L. de Weck: 16.684 Space System Product Development, Spring 2001 Department of Aeronautics & Astronautics, Massachusetts Institute of Technology,AOCS hardware: gyro,Rate Gyros (Gyroscopes) Measure the
24、 angular rate of a spacecraft relative to inertial space Need at least three. Usually use more for redundancy. Can integrate to get angle. However, DC bias errors in electronics will cause the output of the integrator to ramp and eventually saturate (drift) Thus, need inertial update Mechanical gyro
25、s (accurate, heavy); Fiber Optic Gyro (FOG); MEMS-gyros,AOCS hardware: MAG & MTQ,Sensor: magnetometer (MAG) Sensitive to field from spacecraft (electronics), mounted on boom Get attitude information by comparing measured B to modeled B Tilted dipole model of earths field Actuator: torque rod (MTQ) O
26、ften used for Low Earth Orbit (LEO) satellites Useful for initial acquisition maneuvers Commonly use for momentum desaturation (“dumping”) in reaction wheel systems May cause harmful influence on star trackers,AOCS hardware: reaction wheel,One creates torques on a spacecraft by creating equal but op
27、posite torques on Reaction Wheels (flywheels on motors). For three-axes of torque, three wheels are necessary. Usually use four wheels for redundancy (use wheel speed biasing equation) If external torques exist, wheels will angularly accelerate to counteract these torques. They will eventually reach
28、 an RPM limit (3000-6000 RPM) at which time they must bedesaturated. Static & dynamic imbalances can induce vibrations (mount on isolators) jitter. Usually operate around some nominal spin rate to avoid stiction effects: avoid zero crossing.,AOCS hardware: thruster,Thrust can be used to control atti
29、tude but at the cost of consuming fuel Calculate required fuel using “Rocket Equation” Advances in micro-propulsion make this approach more feasible. Typically want Isp 1000 sec Use consumables such as Cold Gas (Freon, N2) or Hydrazine (N2H4) Must be ON/OFF operated; proportional control usually not
30、 feasible: pulse width modulation (PWM) Redundancy usually required, makes the system more complex and expensive Fast, powerful Often introduces attitude/translation coupling Standard equipment on manned spacecraft May be used to “unload”accumulated angular momentum on reaction-wheel controlled spac
31、ecraft.,Attitude Determination,IMU (Inertial Measurement unit): gyro + accelerometer IRU (Inertial Reference Unit): gyro Gyro-stellar system: gyro + star tracker Gyro: rate output, but has bias need to estimate gyro drift: Kalman filter or a low pass filter; integrate to obtain angle. Star tracker:
32、angle output differentiate to obtain rate (will induce noise),Orbit Determination,GPS receiver: obtain measurements Extended Kalman Filter: State propagation (or prediction) (Cowells formulation: nonlinear) State residual covariance matrix propagation (linearized state transition matrix) Kalman gain
33、 computation: steady-state gain can be solved from Riccati equation. State update (or correction) State residual covariance matrix update (or correction): use Joseph form to maintain symmetry.,Attitude control: some definitions,Attitude Control,The environmental effects Spin stabilization Dual-spin
34、stabilization Three-axis active control Momentum exchange systems Passive gravity gradient stabilization,Attitude control systems comparisons,Control actuator torque values,A comparison of various control systems,Attitude control: environmental effects,Solar radiation pressure Force Torque: induced
35、by CM CD=3 for a cylinder. Only a factor for LEO.,Spin stabilization: Requires stable inertia ratio: IzIx=Iy Requires nutation damper: ball-in-tube, viscous ring, active damping Requires torquers to control precession (spin axis drift) magnetically or with jets Inertially oriented Dual-spin stabiliz
36、ation Two bodies rotating at different rates about a common axis Behave like simple spinners, but part is despun (antenna, sensor) Requires torquers for momentum control and nutation dampers for stability Allows relaxation of majar axis rule,Attitude control: spin/dual-spin stabilization,Attitude co
37、ntrol: three-axis active control,Reaction wheels most common actuators Fast; continuous feedback control Moving parts Internal torque only; external still need “momentum dumping” (off-loading) Relatively high power, weight, cost Control logic simple for independent axes,Attitude control: gravity gra
38、dient stabilization,Requires stable inertias: IzIx, Iy Requires libration dampers: hysteresis rods Requires no torquers Earth oriented,courtesy from Oliver L. de Weck: 16.684 Space System Product Development, Spring 2001 Department of Aeronautics & Astronautics, Massachusetts Institute of Technology
39、,Attitude control: momentum exchange,Reaction wheel (RW) systems Momentum bias systems: a single RW is aligned along the pitch axis of the spacecraft which is oriented along the normal to the orbital plane. Control moment gyro (CMG) systems Single gimbal CMG Double gimbal CMG RW has smaller output t
40、han CMG; CMG has singularity in momentum envelope.,AOCS Design: spin stabilization, Suppose there is no thurster on the spacecraft. Maintain current and accurate properties of the spacecraft and alternate configurations. Determine the mass and balance of the spacecraft. Provide adequate gyroscopic “
41、stiffness” to prevent significant disturbance of the angular momentum vector. Approximate rule of thumb: 1.05Ispin/Itrans0.95 Keep track of inertia ratios and the location of the center of mass. Consider nutation, spin-axis orientation, spin rate, and attitude perturbation.,AOCS Design: dual-spin st
42、abilization,Energy dissipation of the spacecraft should be managed. The center of mass of the spinner should be as close to the bearing axis as possible. The bearing axis should be the principal axis of the spinning part to prevent forced oscillations and nutation due to center of mass offset and cr
43、oss products of inertia of the spinner. The spinner should be dynamically symmetric (equal transverse moments of inertia) or the stability of spin about the minor principal axis of the spacecraft must be reevaluated by numerical simulation. For the general case when the despun body has significant c
44、ross products of inertia or if its center of mass is not on the bearing axis, simulation of the system equations is recommended.,AOCS Design: three-axis active control,Ensure that all closed loop control systems exhibit acceptable transient response. Control system torque capability must be sufficie
45、ntly large to correct initial condition errors and maintain attitude limits within specified values in the presence of the maximum environmental disturbances. The control logic must be consistent with the minimum impulse size and lifetime specification of the thrusters. Evaluate system performance i
46、ncorporating as many hardware elements in a simulation as possible, Combine the normal tolerances statistically with the beginning and end-of-life center of mass location and moment of inertia characteristics.,FormoSat-2 follow-on animation,ASH (Acquisition Safe Hold) mode animation: one-axis stabilization SLO (Sun LOck)/STR (Sun TRack) Normal mode animation: (active three-axis active attitude control) GAP (Geocentric Attitude Pointing) MAN (MANeuver) FIP (FIne Pointing) SUP (SUn Pointing) GAP,Questions ?,Dr. 游懷祖 hyounspo.org.tw (03) 578-4208 ext 2286Thank you !,
